LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with

$$\|v_i\|\leq \gamma_{i,n} det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$.

1. Are there lattices where this cannot be improved to $\|v_i\|\leq \gamma_{i,n} det(\Lambda)^{1/(n-i+2)}$?

2. In general are there algorithms (possibly in exponential time) which can guarantee $\|v_i\|\leq \gamma_{i,n} det(\Lambda)^{1/(n-i+2)}$?

LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with

$$\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$.

1. Are there lattices where this cannot be improved to $\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+2)}$?

2. In general are there algorithms (possibly in exponential time) which can guarantee $\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+2)}$?